DYNAMICAL THEORY OF HEAT. 
bo 
(oe) 
Or 
Now, by (27), we have 
dx 
edie a A 4 ; ; : 28), 
dx an, ay _ 
L 
Hence M= an : : : : : : j (80), 
N=c(1—2)+i42—L reas (31). 
55. The expression of the second fundamenta proposition in this case becomes, 
consequently, 
Wn 
Pt thee ty 2), 
which agrees with Carnorv’s original result, and is the formula that has been used 
(referred to above in § 31) for determining by means of REGNAULT’s observa- 
tions on steam. 
56. To express the conclusion derivable from the first fundamental proposi- 
tion, we have, by differentiating the preceding expressions for M and N with 
reference to ¢ and v respectively, 
meee aT eT aa Ay 
dt y—-A' dt (y—-AyP” dt 

dy dn 
oan pat dt\dzx 
do (ie YHA / do 
_fh-e L d (y—”) 
a ae wary} dt 
Hence equation (2) of § 20 becomes 
au +e—h 
Wh tie by eh ap 
Weak ay Moe 
Combining this with the conclusion (32) derived from the second fundamental : 
_ proposition, we obtain 
di Lp 
ed Te ea) 
The former of these equations agrees precisely with one which was first given 
by Cuaustus, and the preceding investigation is substantially the same as the in- 
_ vestigation by which he arrived at it. The second differs from another given by 
_ Cxaustus only in not implying any hypothesis as to the form of Carnot’s func- 
tion, p. 
VOL. XX. PART Il. 44H 

